Jets, flows and Joseph Fourier

Transcription

1 Jets, flows and Joseph Fourier Tom Trainor December, Guy Paić Fest Puebla, Mexico

2 Diverse Phenomena Secret Analogies Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them Joseph Fourier, Trainor

3 λ λ λ λ m=-δ λ λ λ λ φ φ φ Fourier Transforms and Series aperiodic function f bins of width λ F(k) = f periodic, with period λ k k m = m/λ x/λ φ/ F = m dx f(x)e Trainor -ikx = dx f(x + n - / / dx f(x)e -i mx / -im d f( )e - Fourier series coefficients - / / / -ikn = e dx f(x)e - / n=- λ n=- Fourier transform (k - )e m/ -ikx -ik(x +n ) ) cylindrical multipoles m multipole monopole dipole quadrupole sextupole octupole v

4 project ρ/ ρ / ρ ref ref.7. Higher Harmonic Flows - SS D peak project D correlations onto D Fourier series fit to data describes any periodic distribution thin solid curves m = : sextupole + x [(sibling pairs / mixed pairs) ] GeV Au-Au -% Trainor Alver/Roland STAR data m= m= SS Gaussian AS dipole m= SS PRC 8, 595 () < < true baseline fit residuals m= jets AS ZYAM zero sextupole triangular flow (né Mach Cones) (deg)

5 σ σ σ φ φ Higher Harmonics in -5% Au-Au? D Gaussian data AS dipole = SS D peak A D F m (σ φ ) m.8 m = Gaussian width σ φ ( ) = φf F SS AS D Gaussian quadrupole { /} exp -m Fourier coefficients: D Gaussian SS D peak - - only source of higher harmonics is the SS D peak Trainor m= GeV Au-Au -5% AS dipole m= m= m= SS D peak multipoles A X{SS} F m A D/

6 ρ/ ρ ref What is the Same-side D Peak? GeV p-p GeV central Au-Au elongation on φ - (mini) jets! - (/y t ) dn h / dy t p t -integral p t (GeV/c) 7 η < p t > GeV/c GeV p-p 5 y t pqcd Trainor 6 ρ/ ρ / ref.7. elongation on η - the Soft Ridge but detailed study confirms a jet mechanism ν H AA = /n part ρ AA S NN p-p η < -% GeV Au-Au GLS 5 pqcd y t () jets, flows? persistence of jets is inconvenient for perfect liquid imperative: reinterpret SS D peak as flows glasma flux tubes, Fourier series higher harmonics

7 ρ ρ GeV p-p Two-component Model p t spectrum hard component PRD 7, 6 (6) minimum-bias: no trigger condition.5..5 soft -.5 φ η - - y -.5- z proton fragments (/n s ) (/y t ) dn ch /dy t S (y t ) Trainor 7 ρ / ρ ref y t away y CI t.5.5 ρ/.5.5 H pp /n s ρ GeV/c GeV/c away side y t nˆ ch.5.5 y t BEC jets D n ch parton fragments same side ρ ref ρ/ 6 p t (GeV/c) D ρ/ ρ / ρ ref y t ln {(m + p )/m y } t t t y t soft ρ ref hadron p t hard hard - 5 y t GeV/c parton fragments -

8 ρ / ρ ref GeV Au-Au D Angular Structure data % ρ/ ρ / ref Au-Au GeV - - D model - - ρ / ρ ref p t -integral ρ/ ρ / ρ ref ref 5..5 arxiv: ρ / ρ ref ρ / ρ ref % Trainor true zeros relative to jet structure 8 ρ/ ρ / ρ ref ref D angular correlations ρ/ ρ / ref

9 A A D σ η Minimum-bias Jet (minijet) Properties same-side (SS) D and away-side (AS) D peaks GeV 6 GeV / N-N b = SS ST GLS 5 6 ν ν = N bin / N part AS SS A D parton momentum conservation ρ / ρ ref ρ/ ρ ref arxiv:9.8-5%.7. σ φ AS GeV 6 GeV dipole GLS 5 6 centrality ν.5 GeV.5 GeV 6 GeV - 6 GeV.75.5 SS η φ.5 SS D peak SS sharp single D Gaussian.75 transition no ridge 5 narrows Trainor ν 9 ν ST GLS

10 ε ρ η ρ Nonjet Azimuth Quadrupole v {D} A Q {method}(b) Au-Au..5 {} ~ {EP} {} {D} GeV 6 GeV Pb-Pb 7 GeV {EP}.8 b/b A Q {D} from D model fits arxiv: nucleon coll. low-x glue (b) = dn / ch * d A {EP} A ~ Q (b/b ) nonflow * A {D} A Q {D} - (b) v {D} =.5R N Q bin,opt Trainor A D, A Q.7 RHIC GeV Au-Au NJ. jets A quadrupole D quadrupole.5 A Q {} AGS R A Q {D}.8R SPS. A Q {SS} 5 6 A {} = A {D} + A {SS} Q Q Q Q published v data nonjet s NN (GeV) jet-related ρ / ρ ref nonjet quadrupole distinct from SS, AS jet structure SS D peak ν

11 Higher Harmonic Flows at the LHC? v m {method} v m predictions based on SS D peak and nonjet (NJ) quadrupole prediction for GeV left panel..76 TeV η exclusion v {} v {} v {D} NJ quadrupole. v {SS} GeV Au-Au jets v {} ~ v {EP} η < Pb-Pb < η < arxiv: v {SS} v {SS}. v {SS}. N-N η exclusion < η < ν centrality v m {SS} at.76 TeV similar to GeV nonjet quadrupole v {D} also similar Trainor?

12 ρ Converting Jets to Flows. fit D projections on φ with a Fourier series. interpret each series term as a harmonic flow. attribute flows to conjectured A-A initial-state geometry global momentum conservation m= A D {SS} - A D {AS} difference between two large numbers, jet structure attributed to bulk medium elliptic flow SS multipoles: m= A Q {D} + A Q {SS} triangular flow residuals nonjet A {SS} m= A S {SS} m= A O {SS} (b)v {SS} = F A / X m m D jet-related SS D peak - used to be Mach cones - predicted by SS D peak SS D jet peak is fragmented to become flows Trainor

13 SS D Peak and Jets Applying pqcd to spectra and correlations confirms a jet interpretation for the SS D peak Trainor

14 β β dn ch /dy y y min Fragmentation Functions Jets GeV/c TASSO, GeV p(q) FF ee q(q) e + -e D ee s = s = 9 GeV OPAL y max 6 y max y Q = s (GeV) y = ln {(E + p)/m } y = ln(q/m ) max D (y, y ) D (x,q ) xx max xx n (y ) ch u = (y - y min )/(ymax - y min ) PRD 7, (6) Trainor PRC 8, 9 (9) max (u;p, q) /n(y max ) dn/du u * e + -e jets,, 9 GeV (u;p, q).8 beta distribution model parameters p, q FF data described to statistical limits for any jets relevant to RHIC: more than half of all jet fragments fall below GeV/c! all fragmentation [gluon hadron] is parametrized accurately u

15 σ σ η ε η σ η pqcd Folding Integral FDs y integrand ee 6 8 jet y max y E jet (GeV) D ee 6 D 7 5 pp 6 pp y 6 8 y max ln(e /m ) = NLO y max PRC 8, 9 (9) measured FF ensembles Q = s (GeV) dσ dijet / dp t [mb/(gev/c)] parton spectrum.5 mb Cooper et al. GeV d dp dijet t p t 6 jets: STAR UA GeV y 5 integrand integrate 6 8 /y d n h /dydη p (GeV/c) FD ee 5 6 y d nh dy d FD j ( NSD p )/ t (GeV/c) d dy maxd xx(y, y max ) FF dy integrand Trainor 5 dijet max fragment distributions FDs spectrum hard components H (/y) d n h /dydη p (GeV/c) H pp p-p data FD NN 5 6 y

16 dn ch /dξ p Parton Energy Loss in A-A TASSO GeV OPAL GeV vac GeV GeV MLLA vac med med 6 8 ξ p = ln(p jet /p) BW q =.5 (u;p, q) dn ch /dy GeV GeV vac med modeled in β(u;p,q) by changing q q is single energy-loss parameter PRC 8, 9 (9) (u;p, q) y = ln[(e + p)/m ] N. Borghini and U. A. Wiedemann, hep-ph/568 Trainor q=.5 TASSO GeV OPAL GeV vac med 6 8 energy conserved by construction

17 ν ν (/y) d n h /dydη Hard-component H XX H NN p (GeV/c) pqcd calculation parton fragments PRC 8, 9 (9) GeV NSD p-p power law e 5.7y 5 6 y N-N reference ν H AA = /n part ρ AA S NN Evolution -% p-p pqcd calculation η < arxiv:75 νh AA GeV Au-Au GeV/c GLS 5 SS D q y t () dn ch = S NN (y t ) + AA t part t t n y dy H (y ; fragmentation evolution pqcd: describes A-A spectrum hard-component evolution Trainor 7 )

18 Jet Correlations and Hadron Production ρ (b) j (b) n j (b). - - GeV Au-Au fragment pairs SS D peak volume ( η=,) GLS 6 ν centrality η = Au-Au GLS GeV p-p pqcd dijets 6 ν SS D peak (jets) H AA (b) Au-Au GeV p-p Au-Au 6 ν Trainor 8 ( η=,) p-p ( η=,) PRD 7, 6 (6) jet fragment yield from spectra (/n part ) dn ch /dη GeV Au-Au p-p Au-Au quantitative relation: SS D peak volume pqcd jet cross section spectrum hard-component yields PRC 8, 9 () S NN + ν H AA (b) (/n part ) ρ (b) Glasma flux tubes GLS S NN 6 total hadron yield solid curve: jet correlations points: spectrum integrals soft component / of final state is jet fragments ν

19 Summary D correlations include a monolithic SS D peak The SS peak biases all v m {,EP} data nonflow All higher harmonics are part of the SS peak The SS peak is quantitatively linked to pqcd (jets) Fourier was correct: A secret analogy between diverse phenomena is revealed higher harmonic flows minimum-bias jets Trainor 9